Poisson’s Equation in Electrostatics Jinn-Liang Liu Institute of Computational and Modeling Science, National Tsing Hua University, Hsinchu 300, Taiwan. E-mail:
[email protected] 2007/2/4, 2010, 2011, 2012, 2017 Abstract Poisson’s equation is derived from Coulomb’s law and Gauss’s theorem. It is a par- tial differential equation with broad utility in electrostatics, mechanical engineer- ing, and theoretical physics. It is named after the French mathematician, geometer and physicist Sim´eon-Denis Poisson (1781-1840). Charles Augustin Coulomb (1736- 1806) was a French physicist who discovered an inverse relationship on the force between charges and the square of its distance. Karl Friedrich Gauss (1777-1855) was a German mathematician who also proved the fundamental theorems of algebra and arithmetic. 1 Coulomb’s Law, Electric Field, and Electric Potential Electrostatics is the branch of physics that deals with the forces exerted by a static (i.e. unchanging) electric field upon charged objects [1]. The ba- 19 sic electrical quantity is charge (e = −1.602 × 10− [C] electron charge in coulomb C). In a medium, an isolated charge Q > 0 located at r0 = (x0, y0, z0) produces an electric field E that exerts a force on all other charges. Thus, a charge q > 0 located at a different point r = (x, y, z) experiences a force from Q given by Coulomb’s law [2] as Q r − r0 F = qE = q 2 [N], r = |r − r0| . (1.1) 4πεr |r − r0| A dimension defines some physical characteristics. For example, length [L], mass [M], time [T ], velocity [L/T ], and force [N = ML/T 2] (in newton).